The 4-Color Problem

Michael Ann Franks

Delmar Middle School

Michael_ann_franks@hotmail.com

Mathematics

Grades 6-12

-Geometry

-Reasoning

-Spatial Sense

-Problem Solving

Purpose: In this lesson we will examine the problem of coloring a map with as few colors as possible so that no two borders will be colored the same color. This problem has a long history with mathematicians and was only recently solved by using advanced computer technology. By using their mathematics reasoning skills, students will be able to decide how to color a map using the minimal number of colors.

Materials:

-4 Colored pencils

-Activity sheets

-Overhead sheets

-Story – “The Poor Cartographer” or “The Little Mapmaker”

Time Required: 1-2 days

Lesson Procedure: Students will listen to one of two stories about a mapmaker and his/her difficulties. I will place a copy of worksheet 1 on the overhead and introduce a series of simple maps to the students. The students will each receive a copy of the overhead and be asked to solve each of the small maps by using as few colored pencils as possible to color each in. The students will find that the first two are very easy and can take as few as 2 colors. They quickly find that the next one is not so easy and many find that they have made a mistake and then ask for a new page. I discuss the reasons why they made the mistake and ask the students the differences between the maps and what made the problem change. I also discuss the importance of using a pencil to write the first letter of each color in the space prior to coloring so that this will not happen. When they are finished with this sheet they are usually very excited with the problem and then I pass out two or three different maps to the class so that neighbors do not have the same map. I allow the students about 15 minutes to get themselves going on the activity and discover how to make the problem easier for themselves. Most students that follow directions will complete the first map very quickly. When most of the class is done with the first map I stop them and ask some questions:

· What were the fewest number of colors you were able to use in coloring your map?

· How could you change this map to into one that uses fewer colors?

· How can you figure out which colors to use in coloring your map?

· What makes the map hard to color?

· What advice would you give to someone who was trying to color a map with the fewest possible colors?

· How many colors do you think it will take to color this map?

·

I then hold up another map that is more difficult and the students generally say “OO-oo-oo” or “COOL!”. I pass those out and walk around asking questions about how they are approaching this harder map.

When there are only a few minutes left to complete the activity, I stop the entire class and review what we have done, discuss the problem and it’s many possible solutions, and let the students explain how they found the easiest way to approach the problem.

Explorations and Extensions: Have students create their own 4-color problem by placing their pen on an unlined sheet of paper, and draw a long continuous curve, that crosses over itself as many times as they would like. They may not cross at a previous crossing, or ride along a previous portion of a curve. They must cross completely over. It will turn out that they can color these with only two colors every time! They may also visit the web site www.megamath.com and make their own map problem on line.

www.math.utah.edu/~alfeld/math/4color.html

www.math.ucalgary.ca/~laf/colorful/4colors.html

www.c3.lanl.gov/mega-math/workbk/map/mpbkgd

www.cs.uidaho.edu/~casey931/mega-math/workbk/map/mpprstory.html

Assessment and Evaluation Tools:

· Students should be able to answer the questions that were asked of them during the lesson.

· Students should be able to explain why certain maps needed only two colors when others needed more than that.

· Students will be required to complete a third map as a homework activity proving the fewest colors needed for that map.

Content Standards:

· NCTM Standard 2: Patterns, representations, and models

· NCTM Standard 3: Geometry

· NCTM Standard 6: problem Solving

· NCTM Standard 7: Reasoning and Proof

· NCTM Standard 8: Communication

Personal Comments: This lesson is a very important one! It may seem quite trivial but is on the state test and requires concentration and good reasoning. Students really have to work on their own to figure out the problem. It always amazes me when the entire class is on task for this activity and VERY excited at the same time. I always have plenty of the different maps copied and available for those that want more.

There are plenty of websites available for more information on this material.

April 11, 2003

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